A rigidity result for overdetermined elliptic problems in the plane. - PowerPoint PPT Presentation

The problem The proof A rigidity result for overdetermined elliptic problems in the plane. David Ruiz Departamento de Análisis Matemático, Universidad de Granada Equadiff 2015, Lyon, July 6-10.

The problem The proof The problem We say that a smooth domain Ω ⊂ R n is extremal if the following problem admits a bounded solution :  ∆ u + f ( u ) = 0 in Ω   u > 0 in Ω  (1) u = 0 on ∂ Ω   ∂ u ν = 1 on ∂ Ω .  ∂� Here � ν ( x ) is the interior normal vector to ∂ Ω at x , and f is a Lipschitz function. Extremal domains arise naturally in many different problems: incompressible fluids moving through a a straight pipe, free boundary problems and obstacle problems (the so-called Signorini problem).

The problem The proof The problem We say that a smooth domain Ω ⊂ R n is extremal if the following problem admits a bounded solution :  ∆ u + f ( u ) = 0 in Ω   u > 0 in Ω  (1) u = 0 on ∂ Ω   ∂ u ν = 1 on ∂ Ω .  ∂� Here � ν ( x ) is the interior normal vector to ∂ Ω at x , and f is a Lipschitz function. Extremal domains arise naturally in many different problems: incompressible fluids moving through a a straight pipe, free boundary problems and obstacle problems (the so-called Signorini problem). If Ω is a bounded extremal domain, then it is a ball and u is radially symmetric. J. Serrin, 1971.

The problem The proof The BCN Conjecture In 1997, Berestycki, Caffarelli and Nirenberg proposed the following conjecture: If R n \ Ω is connected, then Ω is either a ball B n , a half-space, a generalized cylinder B k × R n − k , or the complement of one of them.

The problem The proof The BCN Conjecture In 1997, Berestycki, Caffarelli and Nirenberg proposed the following conjecture: If R n \ Ω is connected, then Ω is either a ball B n , a half-space, a generalized cylinder B k × R n − k , or the complement of one of them. This conjecture has been disproved for n ≥ 3 by P . Sicbaldi: he builds extremal domains obtained as a periodic perturbation of a cylinder (for f ( t ) = λ t ). P . Sicbaldi, 2010.

The problem The proof Overdetermined problems and CMC curfaces A formal analogy has been observed between overdetermined problems and CMC surfaces: Extremal domains CMC surfaces Serrin’s result

The problem The proof Overdetermined problems and CMC curfaces A formal analogy has been observed between overdetermined problems and CMC surfaces: Extremal domains CMC surfaces Serrin’s result Alexandrov’s result

The problem The proof Overdetermined problems and CMC curfaces A formal analogy has been observed between overdetermined problems and CMC surfaces: Extremal domains CMC surfaces Serrin’s result Alexandrov’s result Sicbaldi example

The problem The proof Overdetermined problems and CMC curfaces A formal analogy has been observed between overdetermined problems and CMC surfaces: Extremal domains CMC surfaces Serrin’s result Alexandrov’s result Sicbaldi example Delaunay surfaces

The problem The proof Overdetermined problems and CMC curfaces A formal analogy has been observed between overdetermined problems and CMC surfaces: Extremal domains CMC surfaces Serrin’s result Alexandrov’s result Sicbaldi example Delaunay surfaces Other extremal domains have been built for f of Allen-Cahn type, with 1. ∂ Ω close to a dilated catenoid. 2. ∂ Ω close to a dilated Bombieri-De Giorgi-Giusti minimal graph ( n = 9). M. Del Pino, F . Pacard and J. Wei, 2015.

The problem The proof BCN conjecture in dimension 2 There are some previous results on BCN conjecture in dimension 2. If f = 0, a quite complete description of the problem has been given in: M. Traizet, 2014. In the semilinear case, there are some previous results: 1. If n = 2, u monotone along one direction and ∇ u bounded, then Ω is a half-plane. A. Farina and E. Valdinoci, 2010. 2. If n = 2, Ω is contained in a half-plane and ∇ u is bounded, then the BCN conjecture holds. A. Ros and P . Sicbaldi , 2013. 3. If n = 2, ∂ Ω is a graph and f is of Allen-Cahn type, then Ω is a half-plane. K. Wang and J. Wei, preprint.

The problem The proof Our result Theorem If n = 2 and ∂ Ω is connected and unbounded, then Ω is a half-plane. This is joint work with Antonio Ros (U. Granada) and P . Sicbaldi (U. Aix Marseille).

The problem The proof Our result Theorem If n = 2 and ∂ Ω is connected and unbounded, then Ω is a half-plane. This is joint work with Antonio Ros (U. Granada) and P . Sicbaldi (U. Aix Marseille). The only remaining case for BCN conjecture in dimension 2 is that of exterior domains. Some partial results are: A. Aftalion and J. Busca, 1998. W. Reichel, 1997.

The problem The proof Step 1: the curvature of ∂ Ω is bounded This is proved by contradiction, via a blow-up argument. Assume that there exists p n ∈ ∂ Ω with K ( p n ) → ±∞ ; by making translations and dilations we can pass to a limit problem:  ∆ u ∞ = 0 in Ω ∞ ,   in Ω ∞ , u ∞ > 0  (2) u ∞ = 0 on ∂ Ω ∞ ,  ∂ u ∞  ν = 1 on ∂ Ω ∞ .  ∂� Here u ∞ is locally bounded and ∂ Ω ∞ is unbounded, connected and has curvature equal to 1 at the origin.

The problem The proof Step 1: the curvature of ∂ Ω is bounded This is proved by contradiction, via a blow-up argument. Assume that there exists p n ∈ ∂ Ω with K ( p n ) → ±∞ ; by making translations and dilations we can pass to a limit problem:  ∆ u ∞ = 0 in Ω ∞ ,   in Ω ∞ , u ∞ > 0  (2) u ∞ = 0 on ∂ Ω ∞ ,  ∂ u ∞  ν = 1 on ∂ Ω ∞ .  ∂� Here u ∞ is locally bounded and ∂ Ω ∞ is unbounded, connected and has curvature equal to 1 at the origin. By a result of M. Traizet, such domain should be a half-plane, and we get a contradiction. M. Traizet, 2014.

The problem The proof Step 2: if u is monotone, Ω is a half-plane. Standard regularity theory implies that the C 1 ,α norm of u is bounded. In particular, ∇ u is bounded.

The problem The proof Step 2: if u is monotone, Ω is a half-plane. Standard regularity theory implies that the C 1 ,α norm of u is bounded. In particular, ∇ u is bounded. The result of Farina and Valdinoci implies Step 2 if ∂ Ω is C 3 . A. Farina and E. Valdinoci, 2010.

The problem The proof Step 2: if u is monotone, Ω is a half-plane. Standard regularity theory implies that the C 1 ,α norm of u is bounded. In particular, ∇ u is bounded. The result of Farina and Valdinoci implies Step 2 if ∂ Ω is C 3 . A. Farina and E. Valdinoci, 2010. Our proof is different and uses the ideas for proving the De Giorgi Conjecture in dimension 2.

The problem The proof Limit directions Take p n ∈ Ω a diverging sequence, and assume that p n | p n | → s ∈ S 1 . We say that s is a limit direction in Ω . It is a limit direction to the left if p n ∈ ∂ Ω , p n = γ ( t n ) , t n → + ∞ . Analogously we define a limit direction to the right. Limits in W Limits to the right Limits to the left Figura: The limit directions.

The problem The proof Limit directions Take p n ∈ Ω a diverging sequence, and assume that p n | p n | → s ∈ S 1 . We say that s is a limit direction in Ω . It is a limit direction to the left if p n ∈ ∂ Ω , p n = γ ( t n ) , t n → + ∞ . Analogously we define a limit direction to the right. Limits in W q Limits to the right Limits to the left Figura: The limit directions.

The problem The proof Step 3: the case θ < π . In this case we can apply the moving plane technique to show that u is monotone along one direction. But then Ω must be a half-plane.

The problem The proof Step 4: the case θ = π . Here we need to apply a tilted moving plane. This technique (valid only for n = 2) has been used in different frameworks:

The problem The proof Step 4: the case θ = π . Here we need to apply a tilted moving plane. This technique (valid only for n = 2) has been used in different frameworks: 1. For CMC surfaces, in N. J. Korevaar, R. Kusner and B. Solomon, 1989. 2. For elliptic problems in half-planes and strips, in L. Damascelli and B. Sciunzi, 2010. 3. For overdetermined problems in R 2 , in A. Ros and P . Sicbaldi, 2013.

The problem The proof Step 4: the case θ = π . Here we need to apply a tilted moving plane. This technique (valid only for n = 2) has been used in different frameworks: 1. For CMC surfaces, in N. J. Korevaar, R. Kusner and B. Solomon, 1989. 2. For elliptic problems in half-planes and strips, in L. Damascelli and B. Sciunzi, 2010. 3. For overdetermined problems in R 2 , in A. Ros and P . Sicbaldi, 2013.

The problem The proof Step 5: the case θ > π In this case the moving plane method is not of help. The proof uses a different argument, based on finding a contact which contradicts the maximum principle.

The problem The proof Thank you for your attention!